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Math 176 Calculus – Sec. 7.2: Trigonometric Integrals I. Evaluating ∫ sin m x cos n x dx A. Strategy for Evaluating ∫ sin m x cos n x dx ( ) 1. If the power of cosine is odd n = 2k + 1 , save one cosine factor and use cos x = 1 − sin x to express the remaining factors in terms of sine: 2 ∫ sin 2 m x cos 2 k +1 x dx = ∫ sin m x cos 2 k x cos x dx ( ) x (1 − sin x ) = ∫ sin m x cos 2 x cos x dx k = ∫ sin m k 2 cos x dx Then substitute u= sinx. 2. If the power of sine is odd (m = 2k + 1), save one sine factor and use sin 2 x = 1 − cos 2 x to express the remaining factors in terms of cosine: ∫ sin 2 k +1 x cos n x dx = ∫ sin 2 k x cos 2 k x sin x dx ∫ (sin x ) cos x sin xdx = ∫ (1 − cos x ) cos x sin xdx = 2 k 2 n k n Then substitute u= cosx. 3. If the powers of both sine and cosine are odd, either 1 or 2 can be used. 4. If the powers of both sine and cosine are even, use the half-angle identities 1 [1 − cos(2 x )] and cos 2 x = 1 [1 + cos(2 x )] 2 2 1 5. It is sometimes helpful to use the identity sin x cos x = sin 2 x 2 sin 2 x = B. Examples -- Evaluate the following 1. ∫ sin θ cos 4 θ dθ 2. 3. ∫ sin ∫ 3 d sin 2 x cos3 x x dx 4. Find the area bounded by the curve of y = sin 4 (3x ) and the x-axis from x = 0 to x=3. II. Evaluating ∫ tan m x sec n x dx A. Strategy for Evaluating ∫ tan m x sec n x dx 1. If the power of secant is even (n = 2k ) , save a factor of sec x and use 2 sec2 x = 1 + tan2 x to express the remaining factors in terms of tangent: ∫ tan m x sec2k x dx = ∫ tan m x sec2k −2 x sec2 x dx ( ) sec x dx x (1 + tan x ) sec x dx = ∫ tan m x sec 2 x k −1 = ∫ tan m 2 Then substitute u= tanx. 2 k −1 2 2. If the power of tangent is odd (m = 2k + 1) , save a factor of secxtanx and use tan2 x = sec 2 x − 1 to express the remaining factors in terms of secant: ∫ tan 2k +1 x secn x dx = ∫ tan2k x secn−1 x sec x tan x dx ( ) = ∫ (sec x − 1) sec k = ∫ tan2 x secn−1 x sec x tan x dx 2 Then substitute u= secx. B. Examples -- Evaluate the following 1. ∫ tan 6 y dy 6 2. ∫ tan 0 sec3 d k n−1 x sec x tan x dx 3. Find the volume of the solid generated by revolving the region bounded by the curve of ( ) ( ) y = sec2 x 2 tan 2 x 2 , the x-axis and the line x = III. Evaluating 2 about the y-axis. ∫ tan x dx , ∫ cot x dx , ∫ sec x dx , ∫ csc x dx A. Strategy for Evaluating 1. When evaluating ∫ tan x dx , ∫ cot x dx , ∫ sec x dx , ∫ csc x dx ∫ tan x dx or ∫ cot x dx rewrite the integrand in terms of sine and cosine and then use u-substitution. 2. When evaluating ∫ sec x dx multiply the integrand by the form of one, and then use u-substitution. 3. When evaluating ∫ csc x dx multiply the integrand by the form of one, and then use u-substitution. sec x + tan x , sec x + tan x csc x + cot x , csc x + cot x B. Examples 2 1. ∫ cot d 4 2. ∫ sec x dx C. Integral Formulas for tan(x), cot(x), sec(x), csc(x) 1. 2. 3. 4. ∫ tan u du = ∫ cot u du = ∫ secu du = ∫ csc u du = -lncos u + C = lnsec u + C lnsin u + C = -ln cscu + C lnsec u + tan u + C -ln cscu + cot u + C